Influence of the Parameters of a Synchronous Reluctance Motor
Abstract Generated abstract
This paper examines how key electrical and mechanical parameters of a synchronous reluctance motor affect its ability to pull into synchronism under constant line voltage. Using the complete Park-Gorev equations and analog-computer calculations in phase coordinates, the authors evaluate the limiting load torque as a function of stator resistance, rotor inertia, direct-axis reactance, saliency ratio, and starting-winding time constants. The calculations indicate that higher stator resistance and inertia reduce pull-in torque, larger rotor time constants improve synchronization, the saliency ratio has little relative effect, and increasing direct-axis reactance raises the relative pull-in torque approximately linearly. Experimental tests on a 4 kW motor show satisfactory agreement with the computed results and support practical parameter selection for achieving rated-load pull-in.
Full Text
INFLUENCE OF THE PARAMETERS OF A SYNCHRONOUS RELUCTANCE MOTOR ON THE PROCESS OF PULLING INTO SYNCHRONISM
E. V. Kononenko, A. F. Fink
(Presented by the scientific seminar of the Departments of Electrical Machines and General Electrical Engineering)
The broad application of synchronous reluctance motors (SRM) in automated electric drives is possible only if their properties in steady-state and transient regimes are well known. From a review of the literature devoted to studies of transient processes in SRM, it follows that, because of the complexity of the physical phenomena occurring in such motors, their theory has not yet been studied to a sufficient degree. In published works on questions of pulling an SRM into synchronism [1—3 and others], the authors confined themselves to approximate methods of investigation, neglecting transient electromagnetic processes.
The process of synchronization is the principal transient process of an SRM. Starting circuits must provide the best conditions for pulling motors into synchronism, as well as their automatic resynchronization if, under overload or under a decrease in the supply voltage, the SRM falls out of synchronism. Pulling into synchronism is determined by the torque produced by the starting winding (asynchronous torque) and by the synchronous torque, which is due to the difference in reluctance along the \(d\)- and \(q\)-axes.
For practical purposes it is very important to know in advance the permissible limiting load torques, according to the conditions of synchronization, for various other parameters.
In the present work, assuming that the SRM operates at constant line voltage, the influence on the calculated value of the pull-in torque of the following factors is considered: the active resistance of the stator winding, the inertia constant of the rotor, the inductive reactance along the direct axis \(x_d\), the ratio \(\dfrac{x_d}{x_q}\), and the rotor-winding constants \(T_d\) and \(T_q\).
Analysis of the influence of the indicated parameters was carried out using the complete Park—Gorev equations [4], taking into account the generally accepted assumptions. Taking the electromagnetic torque as positive in the motoring mode (and the \(q\)-axis leads the \(d\)-axis by an angle of \(90^\circ\)), these equations, using the per-unit system, may be written in operator form as follows:
\[ p\psi_d - (1 - s)\psi_q + r i_d = -U \sin \Theta, \]
\[ p\psi_q + (1-s)\psi_d + r \cdot i_q = U \cdot \cos \Theta, \]
\[ p\psi_{dr} + r_{rd} \cdot i_{dr} = 0, \]
\[ p\psi_{qr} + r_{rq} \cdot i_{qr} = 0, \]
\[ \omega_s T_j p^2\Theta + \psi_d i_q - \psi_q i_d = M_c, \]
where quantities without additional subscripts refer to the stator windings, and those with the subscript \(r\) to the rotor windings, respectively, along the \(d\) and \(q\) axes.
\(T_j\) — inertia constant;
\(\Theta\) — the angle formed by the transverse axis of the rotor poles with the voltage vector;
\(S\) — slip.
The investigations are based on a method for calculating the process of pulling into synchronism on the ÉMU-10 analog computer, considered in detail in [4].
According to this method, the solution was carried out in phase coordinates. The value of the load angle was taken to be the value of the resisting torque \(M_c\), for which the relative speed of the machine rotor in asynchronous operation reaches zero, and the integral curve passes through an unstable point of the “saddle” type.
The motor will resynchronize if, in the event of a violation of dynamic stability, the torque is less than or equal to the load angle.
In the investigation, two limiting values are determined for the load angle \(M_{\mathrm{in}}\), differing by no more than 0.001 relative units, which satisfy the condition that, at the smaller limiting value of \(M_{\mathrm{in}}\), the motor is pulled into synchronism, while at the larger one it continues to operate in the established asynchronous mode.
It is convenient to study the influence of parameters on the magnitude of the load angle in the plane of two parameters. In this case, the remaining parameters are taken as constant.
As the calculations showed, the magnitude of the active resistance of the stator winding and the inertia constant have a great influence on the magnitude of the load angle of the SRM. Figure 1 presents the dependences
Fig. 1. Influence of active resistance and inertia constant on the magnitude of the load angle.
\[ \frac{M_{\mathrm{in}}}{M_{\mathrm{n}}}=f(r) \]
for various inertia constants \(T_j\), calculated for a SynRM having the following parameters:
\[ x_d=2.33;\qquad x_q=0.45;\qquad x_{rd}=2.31; \]
\[ x_{rq}=0.428,\qquad r_{rd}=0.0557;\qquad r_{rq}=0.0428. \]
As can be seen from the curves, with an increase in the active resistance of the stator winding, the pull-in torque decreases. This is explained by the fact that, as \(r\) increases, the aperiodic and sinusoidal components of the torque in the synchronous mode increase, which leads to a decrease in the maximum torque.
It is also evident from Fig. 1 that, with an increase in the inertia constant, the pull-in torque decreases. Indeed, as the inertia constant increases, pulling into synchronism is possible at smaller slips, since for a motor of the given design the quantity
\[ \frac{S^2T_j}{2} \]
is constant, which, according to the starting characteristic, corresponds to a decrease in the value of the pull-in torque.
In designing a SynRM, the proper choice of the values of the time constants of the starting windings \(T_d\) and \(T_q\) is of great importance. Figure 2 presents the dependences
\[ \frac{M_{\mathrm{in}}}{M_{\mathrm{n}}}=f(T_d), \]
obtained for various inertia constants. The motor parameters remained the same, except that \(r=0.049\) and \(T_q=0.25T_d\).
Fig. 2. Influence of rotor parameters and the inertia constant on the value of the pull-in torque.
As can be seen from the data presented, with an increase in the time constants of the starting windings, the pull-in torque increases. In the absence of windings on the rotor, pulling into synchronism is practically impossible, since the steepness of the starting characteristic decreases, which worsens the conditions for entering synchronism.
The magnitude of the synchronous inductive reactance \(x_d\), which is inversely proportional to the air-gap size, has a great influence on the operation of a SynRM.
Figure 3 presents the dependences \(\dfrac{M_{\mathrm{in}}}{M_{\mathrm{n}}}=f\left(\dfrac{x_d}{x_q}\right)\), constructed for different \(T_j\) and the following constant parameters: \(x_q=0.45\); \(r=0.049\); \(x_{rq}=0.428\); \(r_{rd}=0.0557\); \(r_{rq}=0.0428\). Along the ordinate axis are plotted the values of \(\dfrac{M_{\mathrm{in}}}{M_{\mathrm{n}}}\), calculated for the maximum torques corresponding to each value of \(x_d\). Analysis of the dependences makes it possible to establish that the relative value of the input torque changes only slightly with a change in \(\dfrac{x_d}{x_q}\).
Fig. 3.
The influence on the input torque of the inductive reactance \(x_d\) was also determined for various inertia constants and \(\dfrac{x_d}{x_q}=5=\mathrm{const}\) (Fig. 4).
The parameters remaining constant were:
\(r=0.049\); \(r_{rd}=0.0557\); \(r_{rq}=0.0428\). Along the ordinate axis are plotted the values of \(\dfrac{M_{\mathrm{in}}}{M_{\mathrm{n}}}\), calculated as in the preceding case. Analysis of the curves makes it possible to establish that the relative value \(\dfrac{M_{\mathrm{in}}}{M_{\mathrm{n}}}\) increases linearly with an increase in \(x_d\).
The results of the analytical investigations were checked experimentally on a motor of type SRO-41-4 with a power of 4 kW and 1500 rpm, operating from a network with constant voltage through series-connected active resistances. The tested motor had the following parameters: \(x_d=2.05\); \(x_q=0.48\); \(x''_d=0.15\); \(x''_q=0.24\); \(T_d=54.2\) electrical seconds; \(T_q=11.9\) electrical seconds; \(T_j=0.62\) sec. The dependences \(M_{\mathrm{in}}=f(r)\) were determined experimentally for different inertia constants. The experimental results agree satisfactorily with the calculated ones.
The investigations carried out make it possible to draw the following conclusions:
Fig. 4.
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With an appropriate choice of the parameters, a synchronous reluctance motor can be designed with an pull-in torque equal to the rated torque, which makes it possible to start under the rated load on the shaft.
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The obtained relationships make it possible, with accuracy sufficient for practice, to determine the influence of the parameters of an SRM on the magnitude of the pull-in torque.
REFERENCES
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A. M. Langen. Determination of the maximum permissible pull-in torque of a reluctance motor. VEP, No. 12, 1956.
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John Doyglas F. H. Pull-in Criterion for Reluctance Motors. Applications and Industry, No. 4, July 1960.
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D. P. Kostenko. Pull-in of a synchronous reluctance motor. Bulletin of the Kiev Polytechnic Institute, No. 1, 1964.
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E. V. Kononenko, A. F. Fink. Application of the EMU-10 computer for the study of transient electromechanical processes in synchronous reluctance motors. Izvestia TPI, vol. 160, 1966.